adaptive-koopman-mpc

This repository contains supplementary material to the paper "Adaptive Koopman Model Predictive Control of Simple Serial Robots" (submitted to IROS 2025).

Preprint: https://arxiv.org/abs/2503.17902

Real system experiments

To test the adaptive Koopman Model Predictive Control (KMPC) algorithm, trajectory tracking experiments were conducted on a 1R and a belt-driven 2R robot. The following video shows an experiment on the 2R system, which demonstrates the controllers ability to reject force disturbances.

Controller parameters

The table below shows the MPC weights used in real system experiments for the 1R and 2R robot systems.

Controller	System	$Q = diag(.)$	$Q_f = diag(.)$	$R = diag(.)$	Buffer size
lin. MPC	2R	$[2.5, 1.1, 0.01, 0.01]$	$[5.5, 2.5, 0.01, 0.01]$	$[0.30, 0.30]$	-
adapt. KMPC	2R	$[3.5, 3.0, 0, \ldots, 0]$	$[3.5, 3.0, 0, \ldots, 0]$	$[0.02, 0.02]$	1500
stat. KMPC	2R	$[10.5, 8.45, 0, \ldots, 0]$	$[10.5, 8.45, 0, \ldots, 0]$	$[0.15, 0.15]$	1500
lin. MPC	1R	$[1.1, 0.01]$	$[5.5, 0.01]$	$[0.1]$	-
adapt. KMPC	1R	$[3.0, 0, \ldots, 0]$	$[3.0, 0, \ldots, 0]$	$[1.0]$	1000
stat. KMPC	1R	$[8.45, 0, \ldots, 0]$	$[8.45, 0, \ldots, 0]$	$[0.4]$	1000

These weights determine how much emphasis is placed on tracking of intermediate states ($Q$) and final state ($Q_f$), and applied controls ($R$). They correspond to the unaugmented optimization problem, as denoted in equation (6a) in the paper. After state augmentation and rewriting the optimization problem in condensed form, the overall weight matrices are:

$$\begin{align*} \mathbf{Q} = \begin{bmatrix} Q & 0 & \cdots & 0\\ 0 & Q & \ddots & 0\\ \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & \cdots & Q_f\\ \end{bmatrix}, \mathbf{R} = \begin{bmatrix} R & 0 & \cdots & 0\\ 0 & R & \ddots & 0\\ \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & \cdots & R\\ \end{bmatrix}. \end{align*}$$

The buffer size relates to the number of data points stored within the circular buffer, which is used for the extended dynamic mode decomposition in the Koopman based controllers.

Simulation code

In the paper, experimental comparison between adaptive KMPC, linearization MPC and a further Koopman based controller, denoted static KMPC, is done. This repository contains code implementations of adaptive KMPC and comparative controllers, which allows to simulate tracking control and mimic the hardware experiments under simplified conditions. Variable names in the source files are consistent with the notation used in the paper, such that equations can be easily identified.

Usage

First, install Julia:

$ curl -fsSL https://install.julialang.org | sh

After creating a local copy of the repository and starting Julia, you can add the package to your current environment via the package manager:

julia> ] dev /path/to/local/repository

This will allow you to use the functions exported by this package. Exemplary usage is shown in two Jupyter notebooks, which simulate reference tracking control of 1R system and 2R system, along with a comparison of the controllers.